q-exponential and q-gamma functions. I. q-exponential functionsa)

Abstract

Some properties of the q‐exponential functions, standard and symmetric, are investigated for general complex nonzero q. In particular, for ‖q‖ ≠ 1, the asymptotic behavior of the q‐exponential functions of a complex argument z is studied as z approaches ∞ through an infinite geometric sequence {z_n^(ζ) = qⁿζ}_n∈Z of complex numbers with common ratio q. A generalized q‐exponential function is defined and some of its properties are also discussed for q ≠ 0 (including the singular case of q a root of unity, or, more generally, ‖q‖=1).